Subgroups — #14282391

Order 1920.

552 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65811	C₂	1	1	yes	maximal, characteristic
2	#11751025	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745835	C₂	1	4	yes	maximal, complemented
5	#10600528	—	3	1	no	maximal, Hall
6	C₂×C₃×#709	—	2	1	no	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×#6710	—	2	1	no	—
9	C₂²×C₅×C₃⋊C₈	—	2	1	no	—
10	C₃×#4076	—	2	1	no	—
11	C₃×#4076	C₄	1	2	yes	—
12	C₃×#4021	—	2	1	no	—
13	C₃×#4020	C₄	1	1	yes	characteristic
14	C₃×C₄×D₂₀	—	2	1	no	—
15	C₃×C₄×D₂₀	C₂²	1	2	yes	—
16	C₃×#4015	C₄	1	1	yes	characteristic
17	C₄⋊(C₃⋊#168)	—	2	1	no	—
18	C₄⋊(C₃⋊#168)	C₂²	1	2	yes	—
19	C₄×C₅×C₃⋊C₈	—	2	1	no	—
20	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
21	#71802	—	5	1	no	maximal, Hall
22	#65811	D₃	1	1	yes	characteristic
23	C₂×C₄×C₅×C₈	—	3	1	no	—
24	#65213	—	3	1	no	—
25	#64679	—	3	4	no	—
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂×C₃×C₂²⋊D₅	—	2	2	no	—
30	C₃×#709	—	4	2	no	—
31	C₃×#709	—	4	2	no	—
32	C₂×C₃×D₂₀	—	2	1	no	—
33	C₂×C₃×D₂₀	—	2	1	no	—
34	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₄×D₅	—	2	2	no	—
36	C₂×C₃×Dic₁₀	—	2	1	no	—
37	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
41	#6710	—	4	2	no	—
42	C₂×C₃⋊#168	—	2	2	no	—
43	C₂×C₅×C₃⋊C₈	—	2	4	no	—
44	C₂×C₅×C₃⋊C₈	—	2	2	no	—
45	C₃×#685	—	2	2	no	—
46	C₃×#685	—	2	2	no	—
47	C₃×C₄⋊Dic₅	—	2	1	no	—
48	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
49	C₃×#532	—	5	1	no	—
50	#4924	—	5	1	no	—
51	C₂×C₄×C₃⋊C₈	D₅	1	1	yes	characteristic, complemented
52	D₄⋊(C₃⋊C₈)	—	5	4	no	—
53	C₂×#709	—	2	1	no	—
54	C₂²×C₅×C₈	—	6	1	no	—
55	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
56	#4076	—	2	1	no	—
57	#4076	Dic₃	1	2	yes	—
58	C₂×#681	—	6	1	no	—
59	#4021	—	2	1	no	—
60	#4020	Dic₃	1	1	yes	characteristic
61	C₄×D₂₀	—	2	1	no	—
62	C₄×D₂₀	C₂×D₃	1	2	yes	—
63	#4015	Dic₃	1	1	yes	characteristic
64	C₄×C₅×C₈	—	3	2	no	—
65	C₄×C₅×C₈	—	6	1	no	—
66	C₄⋊#168	—	3	2	no	—
67	C₄⋊#168	—	6	1	no	—
68	#1492	—	15	1	no	Sylow
69	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₃×D₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	2	2	no	—
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
77	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₃×C₂²⋊D₅	—	4	4	no	—
82	C₂×C₃×Dic₅	—	2	2	no	—
83	C₃×D₂₀	—	2	4	no	—
84	C₃×D₂₀	—	4	2	no	—
85	C₃×D₂₀	—	4	2	no	—
86	C₃×C₄×D₅	—	4	2	no	—
87	C₃×C₄×D₅	—	4	2	no	—
88	C₃×Dic₁₀	—	4	2	no	—
89	C₃⋊#168	—	4	2	no	—
90	C₅×C₃⋊C₈	—	2	4	no	—
91	C₂×C₃×#41	—	10	1	no	—
92	C₃×C₄⋊Q₈	—	5	1	no	—
93	C₃×#127	—	5	1	no	—
94	C₃×#124	—	10	1	no	—
95	C₃×#121	—	5	2	no	—
96	C₃×#121	—	10	1	no	—
97	C₃×C₄×D₄	—	5	2	no	—
98	C₃×C₄×D₄	—	10	1	no	—
99	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
100	C₂×#208	—	10	1	no	—
101	C₂²×C₃⋊C₈	—	2	1	no	—
102	C₄⋊(C₃⋊C₈)	—	5	2	no	—
103	C₄⋊(C₃⋊C₈)	—	10	1	no	—
104	C₄×C₃⋊C₈	—	2	1	no	—
105	C₄×C₃⋊C₈	C₂×D₅	1	2	yes	—
106	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
107	C₂²×C₄×C₅	#67	1	1	yes	characteristic
108	C₂²×C₄×C₅	#67	1	1	yes	characteristic
109	C₂×C₂²⋊D₅	—	2	2	no	—
110	#709	—	4	2	no	—
111	#709	—	4	2	no	—
112	C₂×D₂₀	—	2	1	no	—
113	C₂×D₂₀	—	2	1	no	—
114	C₂×D₂₀	C₂×Dic₃	1	2	yes	—
115	C₂×C₄×D₅	—	2	2	no	—
116	C₂×Dic₁₀	—	2	1	no	—
117	C₂×C₅×C₈	—	6	4	no	—
118	C₂×C₅×C₈	—	6	2	no	—
119	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
120	C₄²×C₅	#67	1	1	yes	characteristic
121	C₄²×C₅	#67	1	1	yes	characteristic
122	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
123	#685	—	2	2	no	—
124	#685	—	2	2	no	—
125	C₄⋊Dic₅	—	2	1	no	—
126	C₄⋊Dic₅	C₂×Dic₃	1	2	yes	—
127	#681	—	12	2	no	—
128	C₂×#168	—	6	2	no	—
129	#532	—	5	1	no	—
130	#423	—	15	1	no	—
131	C₂×C₄×C₈	—	3	1	no	—
132	D₄⋊C₈	—	15	4	no	—
133	C₂²×C₃×C₅	—	2	1	no	—
134	C₂²×C₃×C₅	—	2	1	no	—
135	C₂²×C₃×C₅	#102	1	2	yes	—
136	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₃×D₅	—	4	2	no	—
138	C₂×C₃×D₅	—	4	2	no	—
139	C₂×C₃×D₅	—	4	2	no	—
140	C₃×C₄×C₅	—	2	1	no	—
141	C₃×C₄×C₅	—	2	1	no	—
142	C₃×C₄×C₅	#102	1	2	yes	—
143	C₃×C₄×C₅	#104	1	2	yes	—
144	C₃×C₄×C₅	#131	1	2	yes	—
145	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
146	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
147	C₃×Dic₅	—	4	2	no	—
148	C₃×#41	—	20	2	no	—
149	C₃×#41	—	20	2	no	—
150	C₂×C₃×Q₈	—	10	1	no	—
151	C₂×C₃×D₄	—	5	2	no	—
152	C₂×C₃×D₄	—	10	2	no	—
153	C₂×C₃×D₄	—	10	1	no	—
154	C₂×C₃×D₄	—	10	1	no	—
155	C₂²×C₃×C₄	—	10	2	no	—
156	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
157	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
158	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
159	C₃×C₄⋊C₄	—	5	2	no	—
160	C₃×C₄⋊C₄	—	10	1	no	—
161	C₃×C₂²⋊C₄	—	10	2	no	—
162	C₃×C₂²⋊C₄	—	10	2	no	—
163	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
164	C₃×C₄²	D₂₀	1	1	yes	characteristic
165	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
166	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
167	#208	—	20	2	no	—
168	C₂×C₃⋊C₈	—	2	4	no	—
169	C₂×C₃⋊C₈	—	2	2	no	—
170	C₂×C₃⋊C₈	—	10	2	no	—
171	C₂³×C₅	#217	1	1	yes	characteristic
172	C₂²×D₅	—	2	2	no	—
173	C₂×C₄×C₅	—	2	2	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	#213	1	2	yes	—
176	C₂×C₄×C₅	#214	1	2	yes	—
177	C₂×C₄×C₅	#217	1	1	yes	characteristic
178	C₂×C₄×C₅	#217	1	1	yes	characteristic
179	C₂×C₄×C₅	#217	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
183	C₂²⋊D₅	—	4	4	no	—
184	C₂×Dic₅	—	2	2	no	—
185	D₂₀	—	2	4	no	—
186	D₂₀	—	4	2	no	—
187	D₂₀	—	4	2	no	—
188	C₄×D₅	—	4	2	no	—
189	C₄×D₅	—	4	2	no	—
190	Dic₁₀	—	4	2	no	—
191	C₅×C₈	—	6	4	no	—
192	#168	—	12	2	no	—
193	C₂×#41	—	10	1	no	—
194	C₂×M₁₆	—	30	1	no	—
195	C₂²×C₈	—	6	1	no	—
196	C₄⋊Q₈	—	5	1	no	—
197	#127	—	5	1	no	—
198	#124	—	10	1	no	—
199	#121	—	5	2	no	—
200	#121	—	10	1	no	—
201	C₄×D₄	—	5	2	no	—
202	C₄×D₄	—	10	1	no	—
203	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
204	C₄⋊C₈	—	15	2	no	—
205	C₄⋊C₈	—	30	1	no	—
206	C₄×C₈	—	3	2	no	—
207	C₄×C₈	—	6	1	no	—
208	C₂×C₃×C₅	—	2	2	no	—
209	C₂×C₃×C₅	#408	1	1	yes	characteristic
210	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
211	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
212	C₃×D₅	—	8	2	no	—
213	C₂³×C₃	—	10	2	no	—
214	C₂³×C₃	#685	1	1	yes	characteristic
215	C₃×Q₈	—	20	2	no	—
216	C₃×D₄	—	10	4	no	—
217	C₃×D₄	—	20	4	no	—
218	C₃×D₄	—	20	2	no	—
219	C₃×D₄	—	20	2	no	—
220	C₂×C₃×C₄	—	2	2	no	—
221	C₂×C₃×C₄	—	2	2	no	—
222	C₂×C₃×C₄	—	10	2	no	—
223	C₂×C₃×C₄	—	20	2	no	—
224	C₂×C₃×C₄	—	20	2	no	—
225	C₂×C₃×C₄	#677	1	2	yes	—
226	C₂×C₃×C₄	D₄₀	1	2	yes	—
227	C₂×C₃×C₄	#685	1	1	yes	characteristic
228	C₂×C₃×C₄	#685	1	1	yes	characteristic
229	C₂×C₃×C₄	#685	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
233	C₃⋊C₈	—	2	4	no	—
234	C₃⋊C₈	—	20	2	no	—
235	C₂²×C₅	—	2	1	no	—
236	C₂²×C₅	—	2	1	no	—
237	C₂²×C₅	#832	1	2	yes	—
238	C₂²×C₅	C₂×#217	1	1	yes	characteristic
239	C₂×D₅	—	4	2	no	—
240	C₂×D₅	—	4	2	no	—
241	C₂×D₅	—	4	2	no	—
242	C₄×C₅	—	2	1	no	—
243	C₄×C₅	—	2	1	no	—
244	C₄×C₅	#832	1	2	yes	—
245	C₄×C₅	#837	1	2	yes	—
246	C₄×C₅	C₂×#213	1	1	yes	characteristic
247	C₄×C₅	C₂×#214	1	1	yes	characteristic
248	C₄×C₅	#948	1	2	yes	—
249	Dic₅	—	4	2	no	—
250	#41	—	20	2	no	—
251	#41	—	20	2	no	—
252	C₂×Q₈	—	10	1	no	—
253	C₂×D₄	—	5	2	no	—
254	C₂×D₄	—	10	2	no	—
255	C₂×D₄	—	10	1	no	—
256	C₂×D₄	—	10	1	no	—
257	C₂²×C₄	—	10	2	no	—
258	C₂²×C₄	D₅×Dic₃	1	1	yes	characteristic
259	C₂²×C₄	#1201	1	1	yes	characteristic
260	C₂²×C₄	#1202	1	1	yes	characteristic
261	M₁₆	—	60	2	no	—
262	C₂×C₈	—	6	4	no	—
263	C₂×C₈	—	6	2	no	—
264	C₂×C₈	—	30	2	no	—
265	C₄⋊C₄	—	5	2	no	—
266	C₄⋊C₄	—	10	1	no	—
267	C₂²⋊C₄	—	10	2	no	—
268	C₂²⋊C₄	—	10	2	no	—
269	C₄²	D₅×Dic₃	1	1	yes	characteristic
270	C₄²	#1201	1	1	yes	characteristic
271	C₄²	#1202	1	1	yes	characteristic
272	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
273	C₃×C₅	#1492	1	1	yes	characteristic, complemented, Hall
274	C₂²×C₃	—	2	1	no	—
275	C₂²×C₃	—	2	1	no	—
276	C₂²×C₃	—	20	2	no	—
277	C₂²×C₃	—	20	2	no	—
278	C₂²×C₃	—	20	2	no	—
279	C₂²×C₃	#3953	1	2	yes	—
280	C₂²×C₃	C₂×#685	1	1	yes	characteristic
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	20	2	no	—
284	C₃×C₄	#3937	1	2	yes	—
285	C₃×C₄	#3953	1	2	yes	—
286	C₃×C₄	#4047	1	2	yes	—
287	C₃×C₄	C₂×#677	1	1	yes	characteristic
288	C₃×C₄	C₂×D₄₀	1	1	yes	characteristic
289	C₂×C₅	—	2	2	no	—
290	C₂×C₅	C₂×#832	1	1	yes	characteristic
291	C₂×C₅	#5234	1	1	yes	characteristic
292	C₂×C₅	C₂×#837	1	1	yes	characteristic
293	D₅	—	8	2	no	—
294	C₂³	—	10	2	no	—
295	C₂³	#6665	1	1	yes	characteristic
296	Q₈	—	20	2	no	—
297	D₄	—	10	4	no	—
298	D₄	—	20	4	no	—
299	D₄	—	20	2	no	—
300	D₄	—	20	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	10	2	no	—
304	C₂×C₄	—	20	2	no	—
305	C₂×C₄	—	20	2	no	—
306	C₂×C₄	#6653	1	2	yes	—
307	C₂×C₄	#6657	1	2	yes	—
308	C₂×C₄	#6665	1	1	yes	characteristic
309	C₂×C₄	#6665	1	1	yes	characteristic
310	C₂×C₄	#6665	1	1	yes	characteristic
311	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
312	C₂×C₄	C₂×#1201	1	1	yes	characteristic
313	C₂×C₄	C₂×#1202	1	1	yes	characteristic
314	C₈	—	6	4	no	—
315	C₈	—	60	2	no	—
316	C₂×C₃	—	2	2	no	—
317	C₂×C₃	—	40	2	no	—
318	C₂×C₃	C₂×#3937	1	1	yes	characteristic
319	C₂×C₃	#65399	1	1	yes	characteristic
320	C₂×C₃	C₂×#3953	1	1	yes	characteristic
321	C₅	#71802	1	1	yes	characteristic, complemented, Sylow
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	20	2	no	—
325	C₂²	—	20	2	no	—
326	C₂²	—	20	2	no	—
327	C₂²	#90965	1	2	yes	—
328	C₂²	C₂×#6665	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	20	2	no	—
332	C₄	#90965	1	2	yes	—
333	C₄	#90977	1	2	yes	—
334	C₄	#91283	1	2	yes	—
335	C₄	C₂×#6653	1	1	yes	characteristic
336	C₄	C₂×#6657	1	1	yes	characteristic
337	C₃	#10600528	1	1	yes	characteristic, complemented, Sylow
338	C₂	—	2	2	no	—
339	C₂	—	40	2	no	—
340	C₂	#11748342	1	1	yes	characteristic
341	C₂	C₂×#90965	1	1	yes	characteristic
342	C₂	C₂×#90977	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#1202
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₃×#65811	C₂
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1492
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #11751025	C₂	semidirect
3. C₂×C₄×C₅×C₃⋊C₈	C₂	semidirect
4. #11745835	C₂	semidirect
51. C₂×C₄×C₃⋊C₈	D₅	semidirect
273. C₃×C₅	#1492	semidirect
321. C₅	#71802	semidirect
337. C₃	#10600528	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×#65811	2
#11751025	2
C₂×C₄×C₅×C₃⋊C₈	2
#11745835	2
#10600528	3
#71802	5